An Integral Boundary Value Problem of Fractional Differential Equations with a Sign-Changed Parameter in Banach Spaces

Yang, Chen; Guo, Yaru; Zhai, Chengbo

doi:https://doi.org/10.1155/2021/9567931

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Abstract Introduction Preliminaries Examples Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 9567931 | https://doi.org/10.1155/2021/9567931

An Integral Boundary Value Problem of Fractional Differential Equations with a Sign-Changed Parameter in Banach Spaces

Chen Yang,¹Yaru Guo,²and Chengbo Zhai²

Academic Editor: Atila Bueno

Received23 Jul 2020

Revised21 Jan 2021

Accepted28 Jan 2021

Published11 Feb 2021

Abstract

This paper is to investigate the existence and uniqueness of solutions for an integral boundary value problem of new fractional differential equations with a sign-changed parameter in Banach spaces. The main used approach is a recent fixed point theorem of increasing -concave operators defined on ordered sets. In addition, we can present a monotone iterative scheme to approximate the unique solution. In the end, two simple examples are given to illustrate our main results.

1. Introduction

With the intensive development of theory and applications of fractional calculus, fractional differential equations have been paid great interest in many fields and thus different boundary conditions of fractional differential equations have also attracted much attention, see [1–42]. However, we can see that there are very few results reported on solutions for boundary value problems of fractional differential equations in Banach spaces.

In 2010, by using contraction mapping principle, fixed point index theory of completely continuous operator, and so on, Bai [2] obtained the existence results of positive solutions for the following three-point problem involving with fractional derivative:where , is the Riemann–Liouville (R-L for short) fractional derivative, and the function is continuous on .

In 2014, by using Guo–Krasnoselskii fixed point theorem, Cabada and Hamdi [4] obtained the existence of positive solutions for the following fractional differential equations with an integral condition:where , is the R-L fractional derivative, and is a continuous function.

As we know, most of the existing results of solutions for fractional differential equation boundary value problems have been obtained in real space . There exists very few papers studied in abstract spaces, and we can find [3, 9] and others.

In 2018, Chen and Gao [3] discussed the following problem of fractional differential equations in a Banach space :where , is the R-L fractional derivative, is continuous, here is a normal cone in , and denotes the zero element of . They obtained the existence results of positive solutions by using fixed point index theory of condensing mapping.

Motivated by these works, we consider the following new form of fractional differential equation with an integral boundary condition:where , , and are real numbers, is a sign-changed parameter, , is the R-L fractional derivative, is continuous, is continuous, here is a normal cone in , and is the zero element of . From the literature, we know that problem (4) is a new form of fractional differential equations. Different from the previous works, we use a recent fixed point theorem for -concave operators defined in ordered set (see [43]) to study (4), and we obtain the existence and uniqueness of solution, while the uniqueness is not treated in the literature [3]. It should be pointed out that this method is interesting for solving nonlinear differential boundary value problems. In particular, we do not need the existence of upper-lower solutions, which is a critical condition in many articles.

2. Preliminaries

In this section, we state some known facts on concave operators; one can see [43–45].

Let be a Banach space, which has a partial order induced by a cone , i.e., if and only if . For , the notation means that there are and such that . Clearly, is an equivalence relation. For (i.e., and ), define . Clearly, is convex. Let with , and define , that is,

Evidently, .

Next, we list the definition of -concave operator and fixed point theorems that will play a key role in this paper.

Definition 1. (see [43]). Let be a given operator which satisfies, for , and there is such that Then, we call a -concave operator.

Lemma 1. (see [43]). Let be an increasing -concave operator and be normal, . Then, has a unique fixed point in . Take any , putting a sequence , ; then, as .

Lemma 2. (see [44]). Let be an increasing -concave operator and be normal, . Then, has a unique fixed point in . Take any , making a sequence , ; then, as .

3. Main Results

Let be an ordered Banach space, positive cone is normal, is the zero element of , and has an identity element , i.e., . In this section, let , and we apply Lemmas 1 and 2 to study problem (4) in abstract Banach space ; the space of all continuous -value functions on interval with the norm . Then, is an ordered Banach space induced by the cone , and is also a normal cone in .

Lemma 3. (see [3, 4]). Assume and . Then, for every , the problem of fractional linear differential equation in Banach spaceshas a unique solution where

Lemma 4. (see [3, 4]). Let and . The function given by (7) has the following properties:(i) for and (ii) for and (iii), for and

Next, we intend to give some existence and uniqueness results for problem (4). We break our discussion up into two sections. Two cases of interest are given in Section 3.1 for and in Section 3.2 for , respectively. In Section 3.3, we discuss the existence and uniqueness corollaries for problem (4) in real space .

3.1. Investigation for the Case

Let

To prove the main results, we need the following assumptions:(H1) is continuous and is continuous, and for .(H2)For and , there is such that .(H3) with for , and there are two constants such that

Theorem 1. Assume and (H1)–(H3) are satisfied; then, problem (4) has a unique solution in , where are given in (8) and (9). Furthermore, set a sequence byfor any taken , and one has as .

Proof. For , by Lemma 4, one hasObviously, it follows , from and Lemma 4.
As , we have . In the following, we will consider a set . By means of Lemma 3, if problem (4) has a solution , thenSo, for any and , we can define an operator by Hence, is a solution of problem (4) if and only if is a fixed point of .
Now, we show that satisfied the definition of -concave operator. For every , , and from condition (H2), one hasTherefore, we obtainIn view of Definition 1, is a -concave operator.
Next, we prove that is increasing. Since , we get , so there is such that ; thus, we obtainBy condition (H1), is increasing.
The main step is to prove that ; in other words, we prove . By (H1), (H3), and Lemma 4, one hasLetSince , , and from (H1) and (H3), we can easily get . It follows that .
Finally, by using Lemma 1, operator has a unique fixed point in , and then,Consequently, is the unique solution of problem (4) in . Taking any , the sequence , , satisfies as . That is,and as .

3.2. Investigation for the Case

In this section, we can get the unique result by using Lemma 2. And, some other assumptions are listed in the following:(H4) is continuous with for , is continuous.(H5)For , is increasing with respect to the second variable, and there are such that(H6)For , there exists such that

Theorem 2. Assume (H4)–(H6) are satisfied; then, for any taken , problem (4) has a unique positive solution in , where . Moreover, taking any initial value , the sequencesatisfies as .

Proof. For , we define an operator bywith given as in Lemma 3.
From Lemma 3, is the solution of problem (4) if and only if is the fixed point of . Noting that and , it is easy to check that . From conditions (H4) and (H5), it is easy to prove that is increasing.
Next, we show that satisfied the definition of -concave operator. From (H6), for and , we obtainHence, .
In the following, we prove that . Firstly, note that , and from (H4), (H5), and Lemma 4, it follows thatSecondly, for , one hasSince , from (H5), it is easy to prove thatHence . It follows that . Finally, by using Lemma 2, the operator has a unique fixed point in , andConsequently, is the unique solution of problem (4) in . For , the sequence , , satisfies as . That is,satisfies as .

Remark 1. The form of problem (4) is more general. Our method is new to study fractional differential equations in Banach spaces, which guarantees the existence and uniqueness of solutions or positive solutions. We also can make an iteration to approximate the unique solution.

3.3. Direct Corollaries

Let , = , and . We can get some obvious results from Theorems 1 and 2. Set(H1’) is continuous, for , and is continuous.(H2’)For and , there is such that (H3’) for , and there are two constants such that

Corollary 1. Let and . Assume that (H1’)–(H3’) hold; then, for , the boundary value problem has a unique solution in , where are given in (30 and 31). Furthermore, set a sequence for any taken , and one has as .

Corollary 2. Let and . Assume that(H4’) is continuous with for and is continuous(H5’)For , is increasing in the second variable(H6’)For , there exists such that then, for , the boundary value problemhas a unique solution in , where . Furthermore, put a sequence byfor any taken , and one has as .

4. Examples

To illustrate our main results, we present two examples.

Example 1. Consider the following boundary value problem:whereand . After a simple calculation,where is given as Lemma 1. Moreover, with . Then,Furthermore, . We can see that is continuous and increasing in the second variable, andwith , . Thus, conditions (H1’) and (H3’) are satisfied. Obviously, it can be expressed as andFrom the Remark 4 in [43], we obtainwhere ; then, condition (H1’) is satisfied. In view of Corollary 1, problem (37) has a unique solution in . For , letWe have as , .

Example 2. Consider the following boundary value problem:where , and are continuous. Take . It is clear that is continuous and increasing in the second variable, with , , and thus, conditions (H4’) and (H5’) are satisfied. Next, for ,where ; then, condition (H6’) is satisfied. In view of Corollary 2, problem (45) has a unique solution in . For , setAnd we have as , .

5. Conclusion

In this paper, we study an integral boundary value problem (4) with sign-changed parameter in Banach spaces. By taking two functions and and using fixed point theorems of increasing -concave operators defined on ordered set , we establish some new existence and uniqueness criteria for problem (4) dependent on different parameters, which also give some different answers to the same type of fractional differential equations’ boundary value problem in the literature [3]. Our results can guarantee the existence of a unique solution with not supposing the existence of upper-lower solutions. As applications, two good examples are presented to illustrate the main results.

Data Availability

No data were used to support the findings of the study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Acknowledgments

This work was supported by Shanxi Province Science Foundation (201901D111020) and Graduate Science and Technology Innovation Project of Shanxi (2019BY014).

References

B. Ahmad and S. Sivasundaram, “On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order,” Applied Mathematics and Computation, vol. 217, no. 2, pp. 480–487, 2010.
View at: Publisher Site | Google Scholar
Z. Bai, “On positive solutions of a nonlocal fractional boundary value problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 2, pp. 916–924, 2010.
View at: Publisher Site | Google Scholar
P. Chen and Y. Gao, “Positive solutions for a class of nonlinear fractional differential equations with nonlocal boundary value conditions,” Positivity, vol. 22, no. 3, pp. 761–772, 2018.
View at: Publisher Site | Google Scholar
A. Cabada and Z. Hamdi, “Nonlinear fractional differential equations with integral boundary value conditions,” Applied Mathematics and Computation, vol. 228, pp. 251–257, 2014.
View at: Publisher Site | Google Scholar
Y. Wang, L. Liu, and Y. Wu, “Positive solutions for a nonlocal fractional differential equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 11, pp. 3599–3605, 2011.
View at: Publisher Site | Google Scholar
C. Zhai, W. Wang, and H. Li, “A uniqueness method to a new Hadamard fractional differential system with four-point boundary conditions,” Journal of Inequalities and Applications, vol. 2018, p. 1, 2018.
View at: Publisher Site | Google Scholar
C. Zhai and W. Wang, “Properties of positive solutions for m-point fractional differential equations on an infinite interval,” Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, vol. 113, no. 2, pp. 1289–1298, 2019.
View at: Publisher Site | Google Scholar
C. Zhai and J. Ren, “The unique solution for a fractionalq-difference equation with three-point boundary conditions-difference equation with three-point boundary conditions,” Indagationes Mathematicae, vol. 29, no. 3, pp. 948–961, 2018.
View at: Publisher Site | Google Scholar
Z.-W. Lv, J. Liang, and T.-J. Xiao, “Solutions to the Cauchy problem for differential equations in Banach spaces with fractional order,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1303–1311, 2011.
View at: Publisher Site | Google Scholar
B. Ahmad, A. Alsaedi, A. Alsaedi, and S. K. Ntouyas, “Fractional order nonlinear mixed coupled systems with coupled integro-differential boundary conditions,” Journal of Applied Analysis & Computation, vol. 10, no. 3, pp. 892–903, 2020.
View at: Publisher Site | Google Scholar
S. Meng and Y. Cui, “Multiplicity results to a conformable fractional differential equations involving integral boundary condition,” Complexity, vol. 2019, Article ID 8402347, 2019.
View at: Publisher Site | Google Scholar
W. Wang, “Properties of Green’s function and the existence of different types of solutions for nonlinear fractional BVP with a parameter in integral boundary conditions, Bound,” Boundary Value Problems, vol. 2019, p. 76, 2019.
View at: Publisher Site | Google Scholar
Y. Sun and M. Zhao, “Positive solutions for a class of fractional differential equations with integral boundary conditions,” Applied Mathematics Letters, vol. 34, pp. 17–21, 2014.
View at: Publisher Site | Google Scholar
S. K. Ntouyas and H. H. Al-Sulami, “A study of coupled systems of mixed order fractional differential equations and inclusions with coupled integral fractional boundary conditions,” Advances in Difference Equations, vol. 2020, p. 73, 2020.
View at: Publisher Site | Google Scholar
J. Tan, M. Li, and A. Pan, “Uniqueness of iterative positive solution to nonlinear fractional differential equations with negatively perturbed term,” Journal of Function Spaces, vol. 2020, Article ID 5747425, p. 6, 2020.
View at: Publisher Site | Google Scholar
Y. Li, H. Sun, and Q. Zhang, “Existence of solutions to fractional boundary-value problems with a parameter,” Electronic Journal of Differential Equations, vol. 141, pp. 1783–1812, 2013.
View at: Google Scholar
K. Zhao, “Triple positive solutions for two classes of delayed nonlinear fractional FDEs with nonlinear integral boundary value conditions,” Boundary Value Problems, vol. 2015, p. 181, 2015.
View at: Publisher Site | Google Scholar
C. Zhai and J. Ren, “Positive and negative solutions of a boundary value problem for a fractional-difference equation,” Advances in Difference Equations, vol. 2017, p. 82, 2017.
View at: Publisher Site | Google Scholar
X. Li, X. Liu, M. Jia, Y. Li, and S. Zhang, “Existence of positive solutions for integral boundary value problems of fractional differential equations on infinite interval,” Mathematical Methods in the Applied Sciences, vol. 40, no. 6, pp. 1892–1904, 2017.
View at: Publisher Site | Google Scholar
B. Ahmad and J. J. Nieto, “Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions,” Boundary Value Problems, vol. 2009, Article ID 708576, pp. 1–11, 2009.
View at: Publisher Site | Google Scholar
J. Wang, M. Fekan, and Y. Zhou, “A survey on impulsive fractional differential equations,” Fractional Calculus and Applied Analysis, vol. 19, pp. 806–831, 2016.
View at: Publisher Site | Google Scholar
W. Wang and X. Guo, “Eigenvalue problem for fractional differential equations with nonlinear integral and disturbance parameter in boundary conditions, Bound,” Boundary Value Problems, vol. 17, no. 6, pp. 2530–2538, 2016.
View at: Google Scholar
Y. Wang and L. Liu, “Uniqueness and existence of positive solutions for the fractional integro-differential equation, Bound,” Boundary Value Problems, vol. 2017, no. 12, 17 pages, 2017.
View at: Google Scholar
J. Wang, S. Peng, and D. O’Regan, “Local stable manifold of Langevin differential equations with two fractional derivatives,” Advances in Difference Equations, vol. 2017, p. 355, 2017.
View at: Google Scholar
Y. Ding, J. Jiang, D. O’Regan, and J. Xu, “Positive solutions for a system of Hadamard-type fractional differential equations with semipositone nonlinearities,” Complexity, Article ID 9742418, p. 14, 2020.
View at: Publisher Site | Google Scholar
H. Wang and L. Zhang, “The solution for a class of sum operator equation and its application to fractional differential equation boundary value problems, Bound,” Boundary Value Problems, vol. 2015, p. 203, 2015.
View at: Publisher Site | Google Scholar
X. Hao, “Positive solution for singular fractional differential equations involving derivatives,” Advances in Difference Equations, vol. 2016, p. 139, 2016.
View at: Google Scholar
N. Xu and W. Liu, “Iterative solutions for a coupled system of fractional differential-integral equations with two-point boundary conditions,” Applied Mathematics and Computation, vol. 244, pp. 903–911, 2014.
View at: Publisher Site | Google Scholar
C. Yang, C. Zhai, and L. Zhang, “Local uniqueness of positive solutions for a coupled system of fractional differential equations with integral boundary conditions,” Advances in Difference Equations, vol. 2017, p. 282, 2017.
View at: Google Scholar
C. Yang, “Existence and uniqueness of positive solutions for boundary value problems of a fractional differential equation with a parameter,” Hacettepe Journal of Mathematics and Statistics, vol. 44, no. 3, pp. 665–673, 2015.
View at: Google Scholar
B. Zhu, L. Liu, and Y. Wu, “Local and global existence of mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay,” Applied Mathematics Letters, vol. 61, pp. 73–79, 2016.
View at: Publisher Site | Google Scholar
L. Zhang, B. Ahmad, G. Wang, and R. P. Agarwal, “Nonlinear fractional integro-differential equations on unbounded domains in a Banach space,” Journal of Computational and Applied Mathematics, vol. 249, pp. 51–56, 2013.
View at: Publisher Site | Google Scholar
X. Zhang, L. Liu, Y. Wu, and B. Wiwatanapataphee, “Nontrivial solutions for a fractional advection dispersion equation in anomalous diffusion,” Applied Mathematics Letters, vol. 66, pp. 1–8, 2017.
View at: Publisher Site | Google Scholar
X. Zhang, C. Mao, L. Liu, and Y. Wu, “Exact iterative solution for an abstract fractional dynamic system model for bioprocess,” Qualitative Theory of Dynamical Systems, vol. 16, no. 1, pp. 205–222, 2017.
View at: Publisher Site | Google Scholar
X. Zhang, L. Liu, Y. Wu, and B. Wiwatanapataphee, “The spectral analysis for a singular fractional differential equation with a signed measure,” Applied Mathematics and Computation, vol. 257, pp. 252–263, 2015.
View at: Publisher Site | Google Scholar
H. Zhang, Y. Li, and J. Xu, “Positive solutions for a system of fractional integral boundary value problems involving Hadamard-type fractional derivatives,” Complexity, Article ID 2671539, p. 11, 2019.
View at: Publisher Site | Google Scholar
G. Wang, X. Ren, and D. Baleanu, “Maximum principle for Hadamard fractional differential equations involving fractional Laplace operator,” Mathematical Methods in the Applied Sciences, vol. 43, no. 5, pp. 2646–2655, 2020.
View at: Publisher Site | Google Scholar
J. Xu, J. Jiang, and D. O’Regan, “Positive Solutions for a class of p-Laplacian Hadamard fractional-order three-point boundary value problems,” Mathematics, vol. 8, no. 3, 2020.
View at: Publisher Site | Google Scholar
X. Liu and M. Jia, “The method of lower and upper solutions for the general boundary value problems of fractional differential equations with p-Laplacian,” Advances in Difference Equations, vol. 28, 2018.
View at: Google Scholar
L. Shu, X.-B. Shu, and J. Mao, “Approximate controllability and existence of mild solutions for Riemann-Liouville fractional stochastic evolution equations with nonlocal conditions of order 1,” Fractional Calculus and Applied Analysis, vol. 22, no. 4, pp. 1086–1112, 2019.
View at: Publisher Site | Google Scholar
Y. Guo, X. Shu, Y. Li, and F. Xu, “The existence and Hyers-Ulam stability of solution for an impulsive Riemann-Liouville fractional neutral functional stochastic differential equation with infinite delay of order,” Boundary Value Problems, vol. 59, 2019.
View at: Google Scholar
X.-B. Shu and Y. Shi, “A study on the mild solution of impulsive fractional evolution equations,” Applied Mathematics and Computation, vol. 273, pp. 465–476, 2016.
View at: Publisher Site | Google Scholar
C. Zhai and L. Wang, “φ − ( h , e )-concave operators and applications,” Journal of Mathematical Analysis and Applications, vol. 454, no. 2, pp. 571–584, 2017.
View at: Publisher Site | Google Scholar
C. Zhai and F. Wang, “Properties of positive solutions for the operator equation and applications to fractional differential equations with integral boundary conditions,” Advances in Difference Equations, vol. 2015, p. 366, 2015.
View at: Publisher Site | Google Scholar
C. Zhai and J. Ren, “Some properties of sets, fixed point theorems in ordered product spaces and applications to a nonlinear system of fractional differential equations,” Topological Methods in Nonlinear Analysis, vol. 49, no. 2, pp. 625–645, 2017.
View at: Google Scholar

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